If a rolled material has its thickness reduced to a critical level, a further reduction of its thickness promotes the elastic deformation of work rolls and makes any further rolling impossible. This critical thickness is called the minimum rollable thickness, and is defined by the following equation: EQU h.sub.min =3.58.multidot.D.multidot..mu..multidot.km/E (1)
where h.sub.min =minimum rollable thickness (mm), D=roll diameter (mm), .mu.=coefficient of friction between the rolls and the rolled material, km=mean deformation resistance of the rolled material (kgf/mm.sup.2), and E=Young's modulus of the rolls (kgf/mm.sup.2).
The minimum rollable thickness resulting from the mutual contact, or kissing of the upper and lower rolls at the opposite ends of the roll barrels is defined by the equation (2): EQU h.sub.min =(C/8).multidot.P.multidot.(2-lnZ) (2)
where C=16 (1-.nu..sup.2)/.pi.E, Z=(L'.sup.2 /b.sup.2).multidot.(B+b)/(B-b), L'=projected contact length (mm), B=barrel length of the rolls (mm), b=sheet breadth (mm), P=rolling force (kgf), .nu.=Poisson's ratio of the rolls. (See, for example, The Third Edition of Iron & Steel Handbook, III (1) Fundamentals of Rolling-Steel Sheets, Maruzen Publishing Co., page 42.)
According to the equation (1), the minimum rollable thickness is in direct proportion to the roll diameter, while it is in inverse proportion to the Young's modulus of the rolls according to the equations (1) and (2), and it is, therefore, usual practice to employ work rolls having a small diameter and a high Young's modulus for rolling a metal foil to make the minimum rollable thickness smaller, as compared with the rolls which are usually employed for cold rolling (to make a sheet having a thickness of, say, 0.2 mm or larger). Examples of the work rolls having a high Young's modulus are ceramic and ultrahard alloy rolls. (See, for example, "Plasticity and Working", Vol. 2, No. 9, page 325 to 334, or Vol. 9, No. 84, page 20 to 29.)
The rolling force per unit width, p (kgf/mm), is expressed by the following equation: EQU p=km.multidot.(R'.multidot..DELTA.h)1/2.multidot.Qp (3)
where Qp is the rolling force function, and R' is the flattened roll radius (mm) as expressed by the following Hitchcock's equation: EQU R'=R.multidot.(1+C.multidot.p/.DELTA.h) (4)
where R=roll radius (mm), and .DELTA.h=reduced thickness (sheet thickness on the inlet side or before rolls, hi-thickness on the outlet side or therafter, h.sub.0) (mm). (See, for example, The Third Edition of Iron & Steel Handbook, III (1) Fundamentals of Rolling-Steel Sheets, Maruzen Publishing Co., page 41.)
As C in the equation (4) is the decreasing function of E, the rolls having a higher Young's modulus E have a smaller flattened radius R', and are also less bent. If the rolls are not satisfactorily flattened or bent for absorbing the factors having an adverse effect on the shape of a product (e.g. lack of uniformity in rolling pressure along the sheet breadth, and its variation with time), it is likely that a product having a defective shape may be obtained. Therefore, Japanese Patent Application Laid-Open No. Hei 1-197004(1989), for example, proposes the use of work rolls having a Young's modulus of 31,000 to 54,000 kgf/mm.sup.2 for the last pass in the manufacture of a metal foil by continuous rolling.
The use of rolls having an upper limit on their Young's modulus as proposed is, however, a disadvantage when it is desirable to decrease the number of passes between rolls and thereby achieve an improved rolling efficiency. The decrease in number of passes necessarily calls for an increase in reduction of thickness per pass and thereby an elevated rolling pressure.
As it is obvious from the equation (2) that the minimum rollable thickness, h.sub.min, resulting from the kissing of rolls is in direct proportion to the rolling pressure and in inverse proportion to the Young's modulus of the rolls, it is limited by the maximum Young's modulus of the rolls if the rolling pressure is raised to the extent allowed by the mill capacity, or the yield point of the rolls, and it is impossible to obtain a metal foil having a smaller thickness. If the Young's modulus of the rolls has an upper limit, the reduction of thickness per pass has its own upper limit which makes it difficult to decrease the number of passes and thereby achieve a high rolling efficiency.
Japanese Patent Application Laid-Open No. Hei 10-34205(1998) proposes that work rolls having a Young's modulus exceeding 54,000 kgf/mm.sup.2 be employed for carrying out at least the last pass with a reduction in thickness of 30% or less when manufacturing a cold rolled metal foil having a thicness of 0.2 mm or less. The use of such hard rolls as have a Young's modulus exceeding 54,000 kgf/mm.sup.2 is, however, likely to result in a rolled product having an irregular shape which is difficult to rectify satisfactorily.
It is, therefore, an object of this invention to provide a process which can manufacture a metal sheet, and particularly a metal foil by rolling with a high efficiency, while not allowing any product having a defective shape to be made.